For me it kind of went in stages.
The first was when I was 17. I went to Best Buy without any specific album in mind (I'd just bought a CD player for my car), and stopped when I saw the Joshua Tree. I'd heard U2 on the radio before (who hasn't?) and thought that they were pretty good, so I took a chance and bought the album. Listening to ISHFWILF on the way home, it hit my like a ton of bricks. I'd heard the song before, but for some reason it really clicked with me then. It's still one of my favorites to this day. I thought RTSS was really amazing as well.
Fast-forward a year or so. I saw U2 on the VH1 My Music Awards performing Beautiful Day. Again, I'd heard the song before, but naturally the performance was amazing. I got ATYCLB for Christmas.
Five years after that, I had JT, ATYCLB, and HTDAAB. I went to see U2 live. Even though I only had three albums, I was shocked that I knew nearly every song they played. One of the few I didn't know was the closer -- Bad. It completely blew my mind. It was a nine-minute version and I didn't want it to end. U2 became my favorite band at that point, and I decided right then and there that I'd go out and buy all of their albums. Bad is still my favorite song.
The next morning I'm driving to work, and one of our radio stations was doing an all-U2 weekend. They played TTTYAATW. Me: "THIS is U2? Whoa." I wasn't expecting something that sounded like that, because I was pretty ignorant of their 90s work other than One and Mysterious Ways. I got off work, went out and bought Achtung Baby, and thus began the obsession.